14 LOCALLY CONVEX SPACES 69 (EVn) For all V e 93 and all A ^ 0 in R (resp. C), we have AV e SB. (EVm) For each V e $8, there exists W e SB such that W + W <= V. Show that there exists a unique topology on E, compatible with the vector space structure, for which 58 is a fundamental system of neighborhoods of 0. 2. Let E be a vector space over R with a denumerable infinite basis (>„), and let % be the set of all balanced absorbing sets in E. Show that 5E does not satisfy axiom (EVm) of Problem 1. (For each n > 0, let An be the set of points ]T tt et such that \tt\ < I In for l=*l 1 Iz-i^n* anc* let A be the union of the sets An . Show that there exists no set M e % such that M + M c A.) 3. Let E be the vector space #R(R) of continuous mappings of R into R. For each continuous function m such that m(t) > 0 for all / e R, let Vm denote the set of functions /e E such that |/(/)| g m(t) for all t e R. (a) Show that the Vm form a fundamental system of neighborhoods of 0 in a topology ^"0 on E which is compatible with the additive group structure of E, but not with the vector space structure. (b) Let D be the vector subspace of E consisting of functions with compact support (12.6). Show that the topology f induced by ^~0 on D is compatible with the vector space structure of D, and is not metrizable. 4. Let (£«)«„! be any family of topological vector spaces, and E = I"I E« the product vector space. Show that the product of the topologies of the E« (1 2.5) is compatible with the vector space structure of E. If each Ea is locally convex and its topology is defined by a family of seminorms CpcuXieL,,, then the topology of E is locally convex and is defined by the family of seminorms /?aA ° pr« (for all a e I and all A e L^), In particular, every (finite or) denumerable product of Fre"chet spaces is a Fre"chet space. 5. Let p be a seminorm on a (real or complex) vector space E. Show that the set N of x e E such that p(x) === 0 is a vector subspace of E. The seminorm p defined by (12.14.8.1 ) on E/N is a norm. Let E be a Hausdorff locally convex space whose topology is defined by a family (/?«) of seminorms, and for each a let Na be the subspace of E consisting of those x e E for which px(x) = 0. Show that E is isomorphic to a subspace of the product of the normed spaces E/N« (the norm on E/N« being p*). In particular, every Fre"chet space is isomorphic to a closed subspace of a denumerable product of normed spaces. 6. A subset A of a vector space E over R is said to absorb a subset B of E if there exists a > 0 such that AB c A whenever |A| :g a. If E is a topological vector space, a subset B of E is said to be bounded (with respect to the topological vector space structure of E) if every neighborhood of 0 absorbs B. (a) The closure of a bounded set is bounded. Every finite set is bounded. If A, B are bounded, then so are A u B and A -f- B. If E is metrizable, every precompact set in E is bounded. (b) Suppose that E is Hausdorff. Then a subset B of E is bounded if and only if, for every sequence (xn) in B and every sequence (An) of real numbers tending to zero, the sequence (Xnxn) tends to 0 in E. (c) Let (E«) be a family of topological vector spaces and E ~ J"I E« their product a (Problem 4). Show that a subset B of E is bounded if and only if pra(B) is bounded in E« for each a.each 1] into G is continuous. Deduce that this mapping can be extended to a non-